Rational S-functions for second order ODEs with non-Liouvillian first integrals
Abstract
The S-function method for rational second order ODEs y''=φ(x,y,y') uses S=Iy/Iz, attached to a first integral I, which satisfies a Riccati equation along the flow; it was designed for Liouvillian I. We study the case of rational S with non-Liouvillian I. From any rational g(x,y,z) with gz≠0 and any linear equation A2w''+A1w'+A0w=0 with polynomial coefficients, the substitution w'/w=g(x,y,y') yields a rational 2ODE whose first integral is built from two independent solutions, and S=gy/gz: the Wronskian cancels, so S does not see the linear equation, and one rational S is shared by equations with and without Liouvillian first integrals. Conversely, given (φ,S), membership in this class reduces to a rational first integral of an explicit planar vector field plus a Möbius-invariant Riccati test in x, which recovers the linear equation. A transfer theorem shows that I is Liouvillian if and only if this equation has a Liouvillian solution, so Liouvillianity is decidable on the class by Kovacic's algorithm: a three-variable problem with no a priori degree bound is reduced to two classical ones. On nine pull-back equations of degree up to $15$, given without information on their construction, the procedure recovers the Airy equation and the Bessel equation of order zero.
Related
- Principle of generating critical points of rational functions and Gauss-Fourier conjecture for second-kind zeros
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- Liouvillian first integrals of differential equations: A Galoisian approach
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BibTeX
@misc{duarte2026rational,
title = {{Rational $S$-functions for second order ODEs with non-Liouvillian first integrals}},
author = {L. G. S. Duarte and L. A. C. P. da Mota},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.32769},
eprint = {2610.32769},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.32769},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Rational S-functions for second order ODEs with non-Liouvillian first integrals
AU - L. G. S. Duarte
AU - L. A. C. P. da Mota
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.32769
UR - https://arxiv.org/abs/2610.32769
AB - The S-function method for rational second order ODEs y''=φ(x,y,y') uses S=I_y/I_z, attached to a first integral I, which satisfies a Riccati equation along the flow; it was designed for Liouvillian I. We study the case of rational S with non-Liouvillian I. From any rational g(x,y,z) with g_z≠0 and any linear equation A_2w''+A_1w'+A_0w=0 with polynomial coefficients, the substitution w'/w=g(x,y,y') yields a rational 2ODE whose first integral is built from two independent solutions, and S=g_y/g_z: the Wronskian cancels, so S does not see the linear equation, and one rational S is shared by equations with and without Liouvillian first integrals. Conversely, given (φ,S), membership in this class reduces to a rational first integral of an explicit planar vector field plus a Möbius-invariant Riccati test in x, which recovers the linear equation. A transfer theorem shows that I is Liouvillian if and only if this equation has a Liouvillian solution, so Liouvillianity is decidable on the class by Kovacic's algorithm: a three-variable problem with no a priori degree bound is reduced to two classical ones. On nine pull-back equations of degree up to $15$, given without information on their construction, the procedure recovers the Airy equation and the Bessel equation of order zero.
ER -